Respectable

Depends on the age of the games. Most current AAA games require an 8gb gpu to run them with a good cpu and memory. You should look up the minimum and suggested requirements on the games you want to play and see if your computer meets those.

Prominent

If you plan on playing 2016+ games I would say 4gb is absolutely not enough. That amount of ram is barely enough for windows at this point. One such example would be Rainbow Six Siege. Even though the game is very well optimised at this point, the game still needs lots of both vram and dram to run the game at high settings.

Honorable

Generally, yes, 4 GB VRAM is enough for Nvidia GPUs @ 1080p. My GTX 970 was still doing fine with most modern games, despite its 3.5 GB "handicap" that did result in the occasional stutter in games that filled the buffer.

I dislike the 3 GB Nvidia cards like the 1060 and 1660ti; they have enough power, but not enough VRAM to keep up with all the textures.

Reputable

The 580 is usually a little bit faster than the 1060. BUT....if I would be you, I just would wait for the GTX 1660Ti that will launch very soon. this card will be $270 and is basically expected to be 25% faster than the 1060 (basically perform similar to the GTX 1070). So you would get much more bang for your buck. I think we will see that card very soon. Many stores already listing it. But there is not a official launch date, but the rumors are running hot, so it can't be much longer. This is a Turing chip without the RTX features (without ray tracing and tensor cores)

The 580 is usually a little bit faster than the 1060. BUT....if I would be you, I just would wait for the GTX 1660Ti that will launch very soon. this card will be $270 and is basically expected to be 25% faster than the 1060 (basically perform similar to the GTX 1070). So you would get much more bang for your buck. I think we will see that card very soon. Many stores already listing it. But there is not a official launch date, but the rumors are running hot, so it can't be much longer. This is a Turing chip without the RTX features (without ray tracing and tensor cores)